Nuprl Lemma : qsum-qle
∀[a,b:ℤ]. ∀[E,F:{a..b-} ⟶ ℚ]. Σa ≤ j < b. E[j] ≤ Σa ≤ j < b. F[j] supposing ∀j:ℤ. ((a ≤ j) ⇒ j < b ⇒ (E[j] ≤ F[j]))
Proof
Definitions occuring in Statement :
qsum: Σa ≤ j < b. E[j],
qle: r ≤ s,
rationals: ℚ,
int_seg: {i..j-},
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
int_upper: {i...},
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
true: True,
subtype_rel: A ⊆r B,
squash: ↓T,
uimplies: b supposing a,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
qsum: Σa ≤ j < b. E[j],
rng_sum: rng_sum,
mon_itop: Π lb ≤ i < ub. E[i],
add_grp_of_rng: r↓+gp,
grp_op: *,
pi2: snd(t),
pi1: fst(t),
grp_id: e,
qrng: <ℚ+*>,
rng_plus: +r,
rng_zero: 0,
itop: Π(op,id) lb ≤ i < ub. E[i],
ycomb: Y,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff
Lemmas referenced :
int_le_to_int_upper,
all_wf,
int_seg_wf,
rationals_wf,
le_wf,
less_than_wf,
qle_wf,
lelt_wf,
qsum_wf,
int_upper_wf,
int_upper_ind,
subtract_wf,
qle_reflexivity,
int-subtype-rationals,
squash_wf,
true_wf,
sum_unroll_base_q,
iff_weakening_equal,
int_upper_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
intformle_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
sum_unroll_hi_q,
decidable__le,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_rel_dep_function,
int_seg_subtype,
subtype_rel_self,
grp_op_preserves_le_qorder,
qle_transitivity_qorder,
qadd_wf,
qadd_com,
qle_witness,
lt_int_wf,
bool_wf,
equal-wf-base,
int_subtype_base,
assert_wf,
le_int_wf,
bnot_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
equal_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
sqequalRule,
lambdaEquality,
functionEquality,
setElimination,
rename,
because_Cache,
hypothesis,
intEquality,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
independent_pairFormation,
productElimination,
independent_functionElimination,
instantiate,
natural_numberEquality,
addEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
imageMemberEquality,
baseClosed,
universeEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
isect_memberFormation,
baseApply,
closedConclusion,
equalityElimination
Latex:
\mforall{}[a,b:\mBbbZ{}]. \mforall{}[E,F:\{a..b\msupminus{}\} {}\mrightarrow{} \mBbbQ{}].
\mSigma{}a \mleq{} j < b. E[j] \mleq{} \mSigma{}a \mleq{} j < b. F[j] supposing \mforall{}j:\mBbbZ{}. ((a \mleq{} j) {}\mRightarrow{} j < b {}\mRightarrow{} (E[j] \mleq{} F[j]))
Date html generated:
2018_05_21-PM-11_59_43
Last ObjectModification:
2017_07_26-PM-06_48_55
Theory : rationals
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